In the special case of \(\rho =q\)  the joint pgf simplifies to \(\varphi ({s},{z}) = p^\alpha [1+q(1-{s}-{z})]^{-\alpha }\)  and the joint
distribution of \(X_1,X_2\)  reduces to the negative trinomial distribution
[1]} with pmf
\({\textsf {P}}[X_1=i,X_2=j] &=\frac{\Gamma (\alpha +i+j)}{\Gamma (\alpha )~i!~j!}\left(\frac{1-q}{1+q}\right)^\alpha \,\left(\frac{q}{1+q}\right)^{i+j}\) 
